Natural Transformation Definition Examples A Worked Example

If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism η_X : F(X) → G(X) in D, such that for every morphism f : X → Y in C we have η_Y O F(f) = G(f) O η_X. This equation can conveniently be expressed by the commutative diagram

If one or both of F or G are contravariant the corresponding horizontal arrow is reversed. If η is a natural transformation from F to G, we also write η : F → G. This is also expressed by saying the family of morphisms η_X : F(X) → G(X) is natural in X.

If, for every object X in C, the morphism η_X is an isomorphism in D, then η is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G.

2 Examples

2.1 A worked example

abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp of all groups with group homomorphisms as morphisms. If (G,*) is a group, we define its opposite group (G^op,*^op) as follows: G^op is the same set as G, and the operation *^op is defined by a*^opb = b*a. All multiplications in G^op are thus "turned around". Forming the opposite group becomes a (covariant!) functor from Grp to Grp if we define f^op = f for any group homomorphism f: G → H. Note that f^op is indeed a group homomorphism from G^op to H^op:

To prove this, we need to provide isomorphisms η_G : G → G^op for every group G, such that the above diagram commutes. Set η_G(a) = a^-1. The formulas (ab)^-1 = b^-1 a^-1 and (a^-1)^-1 = a show that η_G is a group homomorphism which is its own inverse. To prove the naturality, we start with a group homomorphism f : G → H and show η_H o f = f^op o η_G, i.e. (f(a))^-1 = f^op(a^-1) for all a in G. This is true since f^op = f and every group homomorphism has the property (f(a))^-1 = f(a^-1).

2.2 Further examples

If K is a field, then for every vector space V over K we have a "natural" injective linear map V → V** from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps form a natural transformation from the identity functor to the double dual functor.

Consider the category Ab of abelian groupAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fos and group homomorphisms. For all abelian groups X, Y and Z we have a group isomorphism

Hom(X, Hom(Y, Z)) → Hom(X⊗Y, Z).

These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab^op × Ab^op × Ab → Ab.

Natural transformations arise frequently in conjunction with adjoint functorsThe existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mat. Indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors come equipped with two natural transformations (generally not isomorphisms) called the unit and counit.

3 Operations with natural transformations

If η : F → G and ε : G → H are natural transformations between functors C → D, then we can compose them to get a natural transformation εη : F → H. This is done componentwise: (εη)_X = ε_Xη_X. This composition of natural transformation is associative, and allows to consider the collection of all functors C → D itself as a category (see below under Functor categories).

A natural transformation η : F → G is a natural isomorphism if and only if there exists a natural transformation ε : G → F such that ηε = 1_G and εη = 1_F (where 1_F : F → F is the natural transformation assigning to every object X the identity morphism on F(X)).

If η : F → G is a natural transformation between functors C → D, and H : D → E is another functor, then we can form the natural transformation Hη : HF → HG by defining (Hη)_X = H(η_X). If on the other hand K : B → C is a functor, the natural transformation ηK : FK → GK is defined by (ηK)_X = η_K(X).

4 Functor categories

If C is any category and I is a small category, we can form the functor category C^I having as objects all functors from I to C and as morphisms the natural transformations between those functors. This is especially useful if I arises from a directed graph. For instance, if I is the category of the directed graph • → •, then C^I has as objects the morphisms of C, and a morphism between φ : U → V and ψ : X → Y in C^I is a pair of morphisms f : U → X and g : V → Y in C such that the "square commutes", i.e. ψ f = g φ.

5 Yoneda lemma

If X is an object of the category C, then the assignment Y |-> Mor_C(X, Y) defines a covariant functor F_X : C → Set. This functor is called representableIn category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i. sets and functions) allowing one t. The natural transformations from a representable functor to an arbitrary functor F : C → Set are completely known and easy to describe; this is the content of the Yoneda lemmaIn mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (a group being a category with just one object). It allows.

6 Historical Notes

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.

The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly, and those of the singular theory, would be isomorphic. But that in itself stated much less than the existence of a natural transformation of the corresponding homology functors.

Category theory

<Hide>

Home > Natural transformation

1 Definition